Elements of Aomoto's generalized hypergeometric functions and a novel perspective on Gauss' hypergeometric differential equation

Abstract

We review Aomoto's generalized hypergeometric functions on Grassmannian spaces Gr(k +1, n+1). Particularly, we clarify integral representations of the generalized hypergeometric functions in terms of twisted homology and cohomology. With an example of the Gr(2, 4) case, we consider in detail Gauss' original hypergeometric functions in Aomoto's framework. This leads us to present a new systematic description of Gauss' hypergeometric differential equation in a form of a first order Fuchsian differential equation.

20 pages, invited chapter contribution to "Advances in Special Functions and Analysis of Differential Equations," CRC Press, (to be) edited by Praveen Agarwal, Ravi P Agarwal and Michael Ruzhansky. Contents are mainly based on some part of Nucl. Phys. B907:107-153, 2016 (arXiv:1512.06476 [hep-th]) by the author